Nonlinear Phase Noise in Fiber Optical Communication

Transcription

1 Nonlinear Phase Noise in Fiber Optical Communication MOHSN NIZ CHUGHTI Communication Systems Group Department of Signals and Systems Chalmers University of Technology Göteborg, Sweden, 009 EX031/009

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3 BSTRCT Electronic compensation of nonlinear phase noise (NLPN) has been analyzed in the thesis work. Performance comparison of two methods to mitigate nonlinear phase noise has been carried out with Quadrature mplitude Modulation (16 QM) for data transmission in fiber optical communication. These methods focus on linear minimum mean square error (MMSE) minimization and optimal nonlinear post compensation. In the previous results, the best system performance is achieved by nonlinear post compensation with uniformly distributed phase 16 QM constellation. The performance of nonlinear post compensation is further improved by optimization of 16-QM constellation which has four shells with uniform distribution of radii. Each shell has four points with uniformly distributed phases. With the proposed constellation the complexity of the receiver remains unchanged and it becomes more immune to nonlinear phase noise at higher transmission powers. The optimum launch power is increased by db with reduction of SER to about With this improvement the transmission distance of the system is also increased from 3000 km to 3540 km at the previous minimum SER of So in terms of transmission distance the system has an improvement of 540 km. iii

4 CKNOWLEDGEMENTS The thesis work was carried out in the Signals and Systems Department, Chalmers University of Technology, under the supervision of Lotfollah Beygi and examination of ssociate Professor Dr. Erik grell. I would like to thank PhD Student Lotfollah Beygi for his encouragement during the entire period of the thesis work to push the work further and achieve new results. Then I would also like to thank ssociate Professor Dr. Erik grell whom has guided me with very important advices not only during the thesis work but also during my entire Masters programme s period. nother important contribution was the guidance from Dr. Mats Sköld from the Photonics laboratory of the Micro Technology and Nano Science Department of Chalmers University of Technology. His help has been very important in understanding the practical aspects of the fiber optical communication systems. Then at last I would like to thank Dr Pontus Johannisson from the Photonics laboratory of the Micro Technology and Nano Science Department of Chalmers University of Technology for his special interest and guidance in the thesis work. iv

5 TBLE OF CONTENTS 1. INTRODUCTION LINER PHSE NOISE MPLIFIED SPONTNEOUS EMISSION NOISE..... LINER PHSE NOISE IN FIBER OPTICL COMMUNICTION LINK SYMBOL ERROR PROBBILITY OF PSK SYSTEM NONLINER PHSE NOISE OPTICL KERR EFFECT SELF PHSE MODULTION SPM IN MPLIFIED FIBER OPTICL COMMUNICTION LINK STTISTICS OF NLPN SYMBOL ERROR PRORBBILTY OF PSK SYSTEM IN NLPN JOINT PDF OF NLPN ND RECEIVED MPLITUDE COMPENSTION OF NLPN DIGITL BCK PROPGTION SPLIT STEP FOURIER METHOD ITERTIVE ND SYMMETRIC SPLIT STEP FOURIER METHOD LINER MMSE BSED COMPENSTION NONLINER MP COMPENSTION SIMULTION RESULTS SYSTEM MODEL DISTRIBUTED MPLIFICTION LUMPED MPLIFICTIONS VERGE POWER DEFINITION BSE BND SIMULTION INTERPRETTION OF SIMULTION MODEL PRE COMPENSTION POST COMPENSTION NONLINER COMPENSTION COMPRISON OF PHSE ND MPLITUDE STNDRD DEVITIONS EQULLY SPCED PHSE CONSTELLTION... 6 v

6 5.1. PERFORMNCE COMPRISON OF NLPN COMENSTION TECHNIQUES FUTURE WORK DISPERSION NLYSIS CODING NLYSIS REFERENCES PPENDIX PPENDIX B vi

7 CHPTER 1 1. INTRODUCTION The evolution of advanced digital modulation schemes such as Quadrature Phase Shift Keying (QPSK) and Quadrature mplitude Modulation (QM) in wireless and land line communications have attracted significant importance as new alternatives to boost the capacity of fiber optical communication systems. But with every new alternative come new limitations. Fiber optical communication channels are limited by linear and nonlinear impairments. Linear impairments include chromatic dispersion (CD) and polarization mode dispersion (PMD). Nonlinear impairments originate from the variation of the refractive index of the optical fiber dependent on the launched power. This phenomenon is called Kerr effect named after the Scottish Physicist John Kerr. Nonlinear impairments include self phase modulation (SPM), cross phase modulation (XPM) and four wave mixing (FWM). Specifically SPM causes the signal travelling through the fiber to change its own phase. In the case of optically amplified fiber optical communication links, noise introduced by erbium doped fiber amplifiers (EDF) causes the self phase modulation to change in a random fashion and is referred to as nonlinear phase noise (NLPN) which has been mathematically analyzed in the initial chapters and 3. Since information in advanced digital modulation formats is encoded in both phase and amplitude, NLPN becomes a dominant source of symbol errors during demodulation at the receiver. Therefore the integration of digital modulation formats in fiber optical communication requires modifications in digital modulation schemes and the compensation of NLPN and other optical impairments at both transmitter and receiver. In recent years the increasing speed of digital signal processors has made possible the compensation of optical impairments in the electronic domain, very cost effective. Thus various signal processing algorithms, both new and existing, are being proposed to compensate optical impairments in the electronic domain. Techniques to mitigate NLPN in electronic domain have been proposed based on minimization of variance of NLPN. Earlier works have been focused entirely on the analysis of NLPN compensation for binary phase shift keying (BPSK) and Quadrature Phase Shift keying (QPSK) modulation studied in chapter 4. In this thesis work performance of a fiber optical communication system with 16 QM modulation and NLPN compensation has been analyzed in last chapter 5 which also contains the discussion of the simulations and the contribution made to previously obtained results. 1

8 CHPTER. LINER PHSE NOISE In every communication system, the channel has impacts on the signal travelling through it. One of the most common impacts is the dditive White Gaussian Noise (WGN) added to the signal. This noise impacts the signal equally in all degrees of freedom in which the signal is defined thus changing the phase and amplitude of the received signal in a random fashion and having a specific distribution..1. MPLIFIED SPONTNEOUS EMISSION NOISE mplified spontaneous emission (SE) noise in the EDFs is due to spontaneous emission of photons generated from inversion of electrons from the Meta stable state to the stable state. These spontaneously generated photons are also amplified within the amplifier and thus act as noise source in the system. The power spectral density of SE noise is given by [1 p ] S ( ν ) = ( G 1) η hν SE o sp o where ν is the central frequency, G is the gain of the amplifier, h is the Planck s constant, o η is the sp spontaneous emission factor defined below η = sp N N N 1, where Nand N 1 are the atomic populations for the excited and ground states. When an optical band pass filter is used before the receiver the power of SE noise is limited to the bandwidth of the filter. signal travelling in fiber optical channels has two states of polarization. SE noise acts in the two polarizations independently thus increasing the power spectral density of SE noise by a factor of two. The power spectral density or the variance of SE noise is given as follows: P = S H ( ν ν ) dν S ν SE SE f o SE o where H is the transfer function of the optical band pass filter and f ν is the effective bandwidth of o the optical filter. The power spectral density is considered to be almost flat with in the bandwidth of the optical filter so SE noise is WGN with mean zero and variance P. SE

9 .. LINER PHSE NOISE IN FIBER OPTICL COMMUNICTION LINK The total received phase of PSK sytems is given by φr = θ o + θ n where φ r is the total is received phase, θois the transmitted phase and θnis the linear phase noise due to SE noise. The distribution of linear phase noise can be approximated by [, p.139] ρ s s sin n p ( ) cos n n ne ρ θ θ θ θ π where ρis s the optical signal to noise ratio (OSNR) defined below ρs = N σ where is the transmitted amplitude, Nis the number of amplifiers in the system and σ is the variance of SE noise from EDFs. The variance of linear phase noise is given as 1 σ θ n ρs. The distribution of phase noise can also be approximated by Gaussian distribution given below by [, p.139] p θ n ρ s ( θn ) exp θ n ρ. s π.3. SYMBOL ERROR PROBBILITY OF PSK SYSTEM The probability of symbol error for an M-ary PSK system is related to linear phase noise by the following expression [3, p.6] P e π / M 1 θn π / M = p dθ n which gives the following expression π Pe Q ρs sin M where M is the modulation level. It is plotted in figure 1 on the next page. 3

10 QPSK 8-PSK 16-PSK 10-4 BER Eb/No (db) Figure 1: Probability of bit error for various modulation schemes. 4

11 CHPTER 3 3. NONLINER PHSE NOISE 3.1. OPTICL KERR EFFECT Optical Kerr effect was discovered by a Scottish physicist in 1875 John Kerr. Optical Kerr effect or C Kerr effect causes the refractive index of the material to change in proportion to the intensity or power of the electromagnetic field. The change in refractive index is as follows [4]: P n = n eff where n is the nonlinear refractive index, P is the launched power and eff is the effective core area of the optical fiber. This change in refractive index is responsible for the signal to change its own phase i.e., self phase modulation. 3.. SELF PHSE MODULTION The mathematical equation governing the propagation of light in optical fibers is the nonlinear Schrödinger equation given as follows [1, p.108]: z i β + = iγ α t where is launched amplitude, z is the distance, t is time, α is the attenuation coefficient, is the nonlinear parameter defined as π n γ = λ where λ o is the central wavelength, nis the nonlinear refractive index, and eff is the effective core area. The solution to the nonlinear Schrödinger equation when β = 0, which governs dispersion, is given below ( z, t) = (0, t) exp[ iφ ( z, t)] where (z,t) is the amplitude of signal pulse as a function of distance and time o eff NL and φnlis given as L φnl = γ (0, t) dz 0 NL eff φ = γ L (0, t) (1) 5

12 where L αz αl eff = = α 0 L e dz N [1 e ]/ N / α is the effective length of a link with N amplifiers spaced at distance of L. The expression (1) indicates that the phase modulation is directly proportional to the instantaneous power of the signal being transmitted. Figure illustrates self phase modulation. The first curve shows pulse amplitude and the second curve shows the deviation of frequency from the central frequency. The second curve is the derivate of the first curve since frequency is time derivate of phase change and phase change is directly proportional to the instantaneous amplitude. This discussion is mathematically summarized as follows [1, p.109]: φ δω( t) = NL = γ Leff (0, t). t t where δω ( t) is the deviation from the central frequency of the system SPM IN MPLIFIED FIBER OPTICL COMMUNICTION LINK In the context of phase modulated and amplified optical communication links the interplay of amplified spontaneous emission noise and Kerr effect causes the received constellation to rotate in a random fashion and is referred as Gorden-Mollenauer effect. [5] For long haul fiber optical communication (FOC) links with cascaded EDFs, the NLPN added into the system, is expressed mathematically as follows [6]: φ NL γ Leff n1 n1 n n1 n n3 n1 n n3 nn = { } () where is the amplitude of the transmitted signals and nk, k = 1,..., N are the noise terms that have chi squared distribution with two degrees of freedom since noise affects both the In-phase and Quadrature components of the signal. The variance of n k is nk = σ. 4 x 10-4 pulse amplitude time in sec x frequency drift in Hz time in sec x 10-8 Figure : Frequency shift δω( t) due to self phase modulation. 6

13 nalysis of the expression () shows that the process of addition of nonlinear phase noise is a nonlinear time dependent process opposite to that of a linear time independent process. This is because once NLPN is added in the system it remains in the system till it reaches the receiver for demodulation. Figure 3 shows the process of evolution of NLPN as a BPSK signal is being propagated through the fiber. Figure 3(a) shows the probability density function (PDF) of scatter plot when the mean nonlinear phase shift is 0.7 radians and figure 3(b) shows the PDF of scatter plot when the mean nonlinear phase shift is 1.5 radians. This shows that as the signal propagates the mean nonlinear phase shift increases as well as the variance of the NLPN also increases STTISTICS OF NLPN The term + n 1 in expression () has a non central chi squared distribution with two degrees of freedom. So the mean and variance of + n 1 is given as follows [6]: 1 + n = + σ σ { + n } = f ( σ ) = 4 σ + 4σ. 4 1 The covariance between the two terms in expression () is given by σ σ σ Cov( + n, + n + n ) = f ( ) = So the total variance of expression () can be expressed as follows: N N σ NL = ( γ Leff ) f ( Kσ ) + ( N K) f ( Kσ ) K= 1 K = 1 (a) (b) Figure 3: PDF of received BPSK signal with NLPN for mean nonlinear phase shift of (a) 0.7 radians after 000 km (b) 1.5 radians after 4000 km for a transmitted power of -5 dbm. 7

14 which by using mathematical induction gives the following expression ( 1)( ) ( 1) ( 1) σφ = N N + γ L NL eff σ N + + N + N + σ 3. The variance of nonlinear phase noise can be related to OSNR by the following relation The mean of NLPN is expressed as follows: σ φnl φnl. 3ρ s ( ) NL N Leff [ N 1 ] N Leff φ = γ + + σ γ. t first glance, from central limit theorem, it seems that nonlinear phase noise should have a Gaussian distribution but this is not the case since the random NLPN at every amplifier stage is correlated to the NLPN in the previous amplifier stages. Expression () can be expressed in form of a matrix as follows [, p ]: φ = + + (3) NL T T N w x x Cx where w = [ N, 1,...,,1] T N and x = [ n1, n,..., n N ] and C is the covariance matrix which is a product of the two matrices defined as C T = MM,where M is as follows: T M = The exact PDF of the NLPN is difficult to compute so it is computed by taking the inverse Fourier transform of the characteristic function of expression (3) which is given by [, p.150] N T 1 jv(v k w) / λ k Ψ φ ( jv) = exp NL K = 1 1 jvσ λk 1 jvσ λk where v T k and λk are eigen vectors and eigen values of the covariance matrix C. The characteristic function is further simplified by expressing NLPN as normalized NLPN given as follows [7]: Ψ ( jv) = sec jv exp[ ρ jv tan jv] φ 8 s

15 ρs = 11 exact gaussian 1 ρs = ρs = 5 PDF Normalized nonlinear phase noise φ in radians Figure 4: PDF of Normalized nonlinear phase noise φ for various OSNRs. where scaling from normalized nonlinear phase noise to nonlinear phase noise is defined as φ NL φnl = φ ρ s where φis the normalized nonlinear phase shift and is the OSNR. comparison between actual and Gaussian approximation of the NLPN is shown in figure 4 which indicates that the exact distribution of NLPN is not Gaussian but very near to it SYMBOL ERROR PRORBBILTY OF PSK SYSTEM IN NLPN The total received phase of PSK sytems is given by [8] φr = θ o + θ n + φnl where θ is the transmitted phase, o θ is the linear phase noise and n φnl total received phase can also be expressed in terms of normalized NLPN. φnl φr = θo + θn + φ. ρ s is the nonlinear phase noise. The Now considering that the transmitted phase θ o is zero then received phase can be expressed as follows: φnl φr = θn + φ. ρ s

16 The characteristic function of linear phase noise can be calculated from π jvθ n ( jv) p ( θ ) e dθ Ψ θn = π θ where p θ ( θ ) is the PDF of linear phase noise defined earlier in section.. The PDF of total received n n phase signal is the inverse Fourier transform of the product of characteristic functions of linear and nonlinear phase noise given by. where Ψφ NL φnl Ψ φ ( jv) = Ψ ( jv) jv r θ Ψ n φnl ρs n n is the characteristic function of nonlinear phase noise defined in section 3.4. The PDF of total received phase is the inverse Fourier transform of the above given by 1 jvφr pφ ( φ ) ( ) r r = Ψφ jv e dφ r r π. The probability of error for a binary phase shift keying (BPSK) system is calculated by following expression n π / φnl Pe = 1 p ( φ ) dφ. π / φnl φ r r r Error Probability p e, PSK <Φ NL > = OSNR ρ s (db) Figure 5: Symbol error rate for BPSK signal as a function of OSNR. ([8]; reprinted with permission) 10

17 Figure 5 shows the plot for probability of bit error as a function of optical signal to noise ratio (OSNR) for various mean nonlinear phase shifts. The plot depicts the increase in error probability as the mean nonlinear phase shift increases JOINT PDF OF NLPN ND RECEIVED MPLITUDE The joint PDF of received phase θ and amplitude r, for a transmitted power P and phase θ o is given as follows [9]: fr ( r, P) 1 jm( θ θo ) fp, θ ( r, θ ) = + Re { C ( ) } o m r e. (4) π π The first expression is the PDF of received amplitude for transmitted amplitude R and power P which is given by Rice distribution as follows: m= 1 ( r + P/ σ ) R(, ) = + o( / σ ) f r P re I r P where C ( r) is defined as m r + αm jmx tan jmx s α m m m sm r sec jmx ρ r s Cm ( r) = e e I sm where I m is the m th -order modified Bessel function of the first kind and x = γ PL ρ s, αm = ρs sec jmx, s m tan jmx =. jmx 11

18 CHPTER 4 4. COMPENSTION OF NLPN 4.1. DIGITL BCK PROPGTION Digital back propagation has been proposed for joint mitigation of SPM and chromatic dispersion [10]. Digital back propagation is based on the nonlinear Schrödinger equation given as follows: iβ α + + = iγ. z t Nonlinear Schrödinger equation is invertible and can be expressed as iβ α = iγ z t = ' ' ( N + D ) z where N and D are defined as ' N = iγ D iβ α. ' = t So the receiver can be designed based on reverse NLSE in which the received signal can be reverse propagated through a filter with opposite signs of, and for the joint mitigation of NLPN and chromatic dispersion. 4.. SPLIT STEP FOURIER METHOD Split step Fourier method (SSFM) is used to simulate self phase modulation and chromatic dispersion in fiber optical communication simultaneously. In the split step Fourier method nonlinearity acts on the signal alone in time domain then Fourier transform is applied to the signal. In the second step, dispersion is applied in the frequency domain by replacing the in nonlinear Schrödinger equation t by iω. Finally the signal is retrieved by taking the inverse Fourier transform of the signal. The overall operation can be expressed as follows [11]: z + h t = F e F e z t 1 hd( iω ) hn (, ) [ [ (, )]] where F is the Fast Fourier transform (FFT) operation and h is the small incremental distance. 1

19 Figure 6: Illustration of symmetric split Fourier method ITERTIVE ND SYMMETRIC SPLIT STEP FOURIER METHOD In this method the signal is propagated through half of the distance `h with dispersion acting on it and then at middle of the distance nonlinearity acts on it and then in the remaining half of the distance dispersion acts on it again depicted in the figure 6. The process is approximated by the following equation [1, p.4-44]: h N ( z) + N ( z + h) h ( z + h, t) = exp D exp exp D ( z, t) In the case of an amplified optical communication link, for each span between amplifiers several iterations, according to the number of divisions `h, have to be performed for the mitigation of nonlinearity and dispersion, leading to a very high complexity NONITERTIVE SYMETRIC SPLIT STEP FOURIER METHOD Simplification of the above mentioned method is done by assuming that nonlinearity is only concentrated at the beginning of each amplifier link hence only one iteration is performed for each span of fiber thus greatly reducing the complexity of the system for mitigation of nonlinear phase noise. [10] The performance of noniterative asymmetric SSFM is to 3 db poor than iterative symmetric SSFM but the computations are fewer as only one iteration has to be performed for every span of the link. SSFM is indeed quite a powerful method to mitigate both nonlinearity and dispersion yet it still has a high computational cost of approximately 10 5 multiplications per symbol per channel. [13] 13

20 4.5. LINER MMSE BSED COMPENSTION The received power in an amplified fiber optical communication link is = which is also the last term in expression () of NLPN and the only quantitative information at receiver about the NLPN added to the signal. Nonlinear phase noise can then be compensated at the receiver by subtracting a phase proportional to from the received signal phase. The compensated phase is φr α P N, where φ r is the received phase,α is the optimal compensating factor that minimizes the variance of residual nonlinear phase noise calculated by dσ ( α ) / dα 0 and is found to be [6] The variance of nonlinear phase noise is given by N + 1 α γ L. eff φnl α P = N σ φ NL φ NL 3ρ s and the variance of residual nonlinear phase noise is given by σ φnl φnl + α P. N 6ρ The reduction in nonlinear phase noise variance is depicted in figure 7 for an SNR of 16.0 db versus increasing mean nonlinear phase shift which is a function of number of spans in the link. Figure 8 on the next page indicates the performance of linear MMSE compensation through the visual analysis of the received signal PDF before and after compensation. s performance of linear MMSE compensation NLPN variance compensated NLPN variance variance of NLPN nonlinear phase shift in radians Figure 7: Performance linear MMSE compensator. 14

21 (a) (b) (c) (d) Figure 8: Received (a),(c) and compensated (b) (d) PDF for transmitted power of -3 dbm after 4000 km and 8000 km. Figure 8 clearly shows that the higher the nonlinear phase noise, the higher is the residual nonlinear phase noise variance after compensation as indicative in figures 8(c) and 8(d). Figure 9 on the next page analyzes the effect of NLPN through another prespective. It shows the plot of phases for transmitted symbols and their PDF. It also shows the the compensated phases and their PDF which indicates that the PDF of phase comes back to zero mean after compensation. Now under the assumption that linear and nonlinear phase noises are independent the total variance of received signal is the sum of both the variances and so the total variance can be expressed as follows: 1 φ = + r θ φ + n NL ρs σ σ σ φ NL 3ρ s and after linear MMSE compensation the total variance can be expressed as follows: φnl σ = σ + σ +. 6ρ 1 φr θ φ n NL + α PN ρs s 15

22 phase in radians nonlinear phase noise NLPN compensated NLPN PDF of phase number of transmitted symbols nonlinear phase noise distribution 6 NLPN PDF compensated NLPN PDF phase in radians Figure 9: Plot and histogram of NLPN for transmitted power of -3dBm and system length of 4000 km. Figure 10 compares the theoretical and simulated variances of the total received phase versus propagation distance expressed in number of amplifier spans. The results indicate an almost match between simulated and theoretical variances before and after compensation of NLPN NLPN variance compensated NLPN variance theoretical NLPN variance theoretical compensated NLPN variance variance of total phase noise Number of spans Figure 10: Theoretical and simulated total phase variance for a system length of 4000 km and transmitted power of -3 dbm. 16

23 4.6. NONLINER MP COMPENSTION Nonlinear compensation compensates the NLPN by subtracting from the received phase an optimal phase θ ( r) that represents the centre phase which satisfies the following condition [9] c f ( r, θ ( r) + π / M) = f ( r, θ ( r) π / M) P, θ c P, θ c where M is the modulation level,, (, ) is the joint PDF of the received amplitude r and phase for a transmitted power P and phase as presented in section 3.6 and ( )± / are the rotated maximum likelihood (ML) boundaries in NLPN. This optimal phase compensator is a nonlinear function of r in the form of ( )= + + and hence a nonlinear compensator. It is also a (maximum a posteriori probability) MP compensator since the constants, and are functions of the transmitted power P. The constants are formulated in ppendix. Figure 11 shows the PDF of a received signal and its ML boundaries for a QPSK signal at an average received power of -4 dbm, for a distance of 5000 km. Figure 1 on the next page shows the PDF of a compensated signal and its ML boundaries. The ML boundaries become straight after compensation which can easily be implemented in the receiver. (a) (b) Figure 11: (a) Received PDF at -4 dbm after 5000 km (b) ML decision boundaries 17

24 (a) (b) Figure 1: (a) Compensated PDF at -4 dbm after 5000 km (b) ML decision boundaries. 18

25 CHPTER 5 5. SIMULTION RESULTS 5.1. SYSTEM MODEL Table 1 lists the parameters used for simulations.the parameters are same as in [9]. γ Nonlinear coefficient 1.W -1 /Km v opt Optical filter bandwidth 4.7 GHz λ Wavelength 1.55 um α ttenuation coefficient 0.5 db/km η sp Spontaneous emission factor 1.41 Table 1: Parameters for simulations. ttenuation in the fiber is completely compensated by the EDFs and the SE noise added to the system is given by σ = S v = hvη v αl which has already been defined in section.1. The sp opt sp opt gain of each amplifier is equal to αl where L is fiber length between each amplifier. 5.. DISTRIBUTED MPLIFICTION In distributed amplifications the signal travelling through the fiber is amplified throughout the fiber length or in other words the system has a uniform amplification throughout the fiber. It can also be assumed as a system with very small incremental lengths between amplifiers and having a large number of amplifiers LUMPED MPLIFICTIONS Lumped amplification is practically implemented in long haul fiber optical communication links. Each amplifier is placed at distances from about 40 to 100 kilometers for 10 Gbps systems. The gain of each amplifier is equal the loss in the span between the EDFs. 19

26 lumped amplification 0.3 instaneous power average power power distributed amplification 0.3 instaneous power average power power distance in km 5.4. VERGE POWER DEFINITION Figure 13: lumped and distributed amplification. The average power travelling through the fiber is given as follows: L 1 αz 1 e P = P0 e dz L = αl 0 α L where is the launched power and is the average power travelling through the fiber. In order to maintain the average received power almost equal to that of launched power distributed amplification is used in simulations. Figure 13 illustrates difference between lumped and distributed amplification in terms of powers travelling through the fiber. In lumped amplification, launched power and average power are different where as in distributed amplification launched power and average powers are almost the same indicated by red line. In [9] distributed amplification model has been used, so the simulations in the thesis work also use distributed amplification BSE BND SIMULTION The simulations were carried out by only considering the base band symbols of the modulated signal. Then NLPN noise was added in steps equal to the number of the amplifiers in the system. In the simulations shot noise and thermal noise [1, p ] from the receiver diodes was ignored. Chromatic dispersion was also ignored in the simulations. 0

27 5.6. INTERPRETTION OF SIMULTION MODEL The first step in the simulations was to confirm whether the simulation results match with the theoretical model or not? Here Gaussian approximation of the exact model presented in earlier chapters about linear phase noise and NLPN is used to confirm the simulation model. The received signal is sum of the linear and nonlinear phase noise given by φr = θ n + φ NL where φris the total received phase, θ nis linear phase noise and φnl received phase, approximated by Gaussian distribution is as follows: p φ r 1 ( φr ) = exp σ φ π r ( φ ) r φnl σ φr is NLPN. The distribution of total Where φ is the mean nonlinear phase shift and total received phase variance is given by NL σ = σ + σ φr φnl θn whereσ φnl andσ θ are the variances of NLPN and linear phase noise. The variances have been quantified n in the chapters 3 and chapter. Figure 14 shows the received signal PDF of a BPSK modulated signal, with an average received power of -5 dbm and system length of 3000 km which has been used to confirm the Gaussian approximation of the received phase. Figure 14: Received signal PDF for a BPSK signal for a system length of 3000 km and average received power of -5 dbm 1

28 6 5 Total phase PDF simulated approximated phase in radians Figure 15: Distribution of total phase of BPSK signal in Figure 14. Figure 15 shows that there is a good match between the Gaussian approximation and simulated results which confirms the correctness of the simulation model. In following sections three methods to compensate NLPN are discussed when 16 QM is used for data transmission PRE COMPENSTION Pre compensation in [9] is implemented by pre rotating the constellation at the transmitter, by the mean nonlinear phase shift φ NL. The pre rotation for each point in a square 16 QM constellation is different since the mean nonlinear phase shift is a function of transmitted amplitude given by φ NL eff N γ L POST COMPENSTION Post compensation in [9] is implemented by linear MMSE compensation along with pre compensation at the transmitter. When pre compensation is used along with post compensation the pre compensation is not φ N γ L NL eff rather the pre compensation is given by φ NL ( Pt + ) γ Leff P γ L rec eff σ = φnl where Pre c is the received power and Pt is the transmitted power and

29 (a) (b) Figure 16: Received signal PDF after (a) pre compensation and (b) post compensation for system length of 3000 km and at received power of -1 dbm. γ L P eff rec = φ αp r rec is the mean of the phase after post compensation given by φr α P terms of transmitted amplitude as follows: φ NL rec. It can be further elaborated in ( + ) Leff γ Leff + n n γ σ N = φnl In other words the mean, φr αprec is subtracted from compensation doesn t over compensate the received signal. φ NL eff N γ L so that the post Figure 16 shows the PDF of a 16 QM signal after pre and post compensation. NLPN is significantly reduced when post compensation is used along with pre compensation, as apparent in figure 16(b) NONLINER COMPENSTION 16 QM constellation has three shells with three different amplitudes. In order to implement nonlinear compensation discussed in section 4.6 for a 16 QM signal, the receiver requires the knowledge of transmitted power. This is so because the constants, and are functions of transmitted powers. Thus at the receiver 16 QM received signal is separated into three amplitudes by ML detection to determine the transmitted powers for each received symbol. fter this post nonlinear compensation is applied to each received signal depending on the estimated value of transmitted the power. 3

30 5.10. COMPRISON OF PHSE ND MPLITUDE STNDRD DEVITIONS s described in section 3.6 the distribution of received amplitudes have Rice distribution which is given by and the variance of is given by where σ is the variance of SE noise and L f r P re I r P ( r + P / σ ) R (, ) = + o ( / ) σ πσ r σ r = σ + r L1/ 1/ σ is the Laguerre polynomial defined as x / x x L1/ ( x) = e (1 x) I0 xi1 where I 0 and I 1 are the modified Bessel functions of first kind of order 0 and 1. From section 4.6 the variance of the total received phase is given by 1 φ = + r θ φ + n NL ρs σ σ σ φ NL 3ρ s comparison of the variances of the received amplitude and total phase is illustrated in figure 17. From the analysis of figure 17 we observe that at low power levels the OSNR is low, so received amplitudes have high standard deviation. s the transmitted power is increased the OSNR increases resulting in the decrease of standard deviation of the received amplitudes. On the other hand, at high OSNRs the NLPN becomes dominant increasing the standard deviation of the received phase Standard deviation of amplitude σ r Standard deviation of phase in degrees σ φr transmitted power in dbm Figure 17 : Comparison of received amplitude and phase standard deviation. 4

31 PDF of amplitudes of 16 QM constellation simulated first shell second shell third shell (a) (b) Figure 18: (a) Received 16 QM constellation at -5dBm. (b) Distribution of received amplitudes of 16 QM Constellation at -5 dbm. Figure 19: Constellation after nonlinear post compensation at -5 dbm. Scatter plot of a 16 QM signal at -5 dbm is shown in figure 18(a). The distribution of amplitudes of the constellation in figure 18(a) is shown in the figure 18(b). From the analysis of figure 18(b) we can observe that the ML detection of the second and third shell amplitudes is erroneous as the PDFs of second and third shell overlap. In such a situation the received signal constellation after nonlinear post compensation is shown in figure 19. We can observe in the scatter plot in figure 19, that some of the points in the second shell are over compensated. This over compensation results in poor performance of nonlinear compensation at low transmitted powers. 5

32 5.11. EQULLY SPCED PHSE CONSTELLTION In a square 16 QM constellation the points in the middle shell are not uniformly distributed as shown in figure 0. This results in unequal decision regions due to nonlinear phase noise. So the performance of nonlinear post compensation can be further improved by uniformly distributing the points in the middle shell of a square 16 QM constellation as shown in figure 0 by dots. Based on earlier discussion a simple conclusion is drawn that by using a multilevel constellation having uniform distribution in terms of amplitude and phase the performance of the nonlinear post compensation can be further improved. Keeping in view the square 16 QM constellation, optimization of 16 QM constellation is done by uniformly distributing all 16 points in four shells instead of three, keeping the lowest and highest shell radii equivalent to that of square 16 QM constellation, which are calculated in appendix B. The modified circular 16 QM constellation along with its ML decision regions is shown in figure 1 on next page. 6 4 Scatter plot Equally spaced 16 QM constellation Orignal 16 QM constellation Quadrature In-Phase Figure 0: Constellation of square and equal phase 16 QM constellation. 6

33 5 Scatter plot Quadrature In-Phase Figure 1: Circular 16 QM constellation and decision regions Scatter plot 5 Scatter plot Quadrature Quadrature In-Phase (a) In-Phase (b) Figure : (a) NLPN in circular 16 QM (b) Compensated circular 16 QM Constellation and decision boundaries. Figure (a) illustrates the received constellation of a circular 16 QM constellation after a propagation distance of 3000 km and average received power of 0 dbm. Figure (b) shows the compensated constellation by nonlinear post compensation. From the analysis of figure (b) it is apparent that noise has higher variance in terms of phase than amplitude so the decision boundaries are close to ML. The advantage of the these approximate ML boundaries is that they can be easily implemented in the receiver if decisions are made in terms of amplitude and phase since they can be considered as straight decision boundaries in terms of amplitude and phase. 7

34 5.1. PERFORMNCE COMPRISON OF NLPN COMENSTION TECHNIQUES Symbel error rate pre compensation with square 16 QM constellation pre and post compensation with square 16 QM constellation nonlinear compensation with square 16 QM constellation nonlinear compensation with equal phase 16 QM constellation nonlinear compensation with circular 16 QM constellation launched power in dbm Figure 3: Performance comparison of compensation techniques Figure 3 compares the performance of compensation techniques for mitigating nonlinear phase noise, for transmission distance of 3000 km and with all constellations normalized at the same transmit energy. With square 16 QM constellation, the optimum point of performance for only pre compensation is at -7 dbm. When linear MMSE post compensation is used along with pre compensation the optimum performance point increases to -4 dbm with symbol error rate (SER) at around s the transmit power increases the NLPN again comes into play further degrading the performance of the system. When nonlinear post compensation is used the optimum point of performance is increased to about -1 dbm with SER reduced to When equal phase 16 QM constellation is used the performance of nonlinear compensation is further improved as the optimum point of performance is increased to around 0 dbm with SER reduced to This improvement is because nonlinear post compensation assumes uniform ML boundaries in NLPN and the uniform distribution of points in terms of phase implements the assumption. 8

35 dbm Symbel error rate dbm nonlinear compensation with equal phase 16 QM constellation nonlinear compensation with circular 16 QM constellation distance in Km Figure 4: SER versus system length with nonlinear compensation. fter this circular 16 QM constellation is tested along with nonlinear post compensation. The results achieved are further improved as the optimum point of the performance is pushed to dbm with SER reduced to around This improvement is due to two reasons. The circular 16 QM constellation has a uniform distribution of phases. Secondly introduction of four shells reduces the amplitude of points in second shell thus reducing the total variance of NLPN of the constellation at higher transmit powers. The key advantage here is that with the same complexity of the receiver, the system becomes more immune to nonlinear phase noise at higher transmitted powers thus enabling the system to transmit data to further distances for the same symbol error probability. Figure 4 compares the performance of nonlinear post compensation versus increasing system lengths at optimum launched powers of 0 dbm, for equal phase 16 QM constellation, and dbm for circular 16 QM constellation. The system length increases from 3000 km to 3540 km for the symbol error rate of 10-6 with circular 16 QM constellation. So the overall advantage in terms of distance is around 540 km for the least SER of 10-6 achieved in [9]. 9

36 CHPTER 6 6. FUTURE WORK 6.1. DISPERSION NLYSIS Chromatic dispersion causes the signal to spread in time causing the amplitude of the signal pulse to decrease. Since nonlinear phase shift is directly proportional to the signal amplitude so the variance of the nonlinear phase shift should also decrease with increase in chromatic dispersion [14]. similar conclusion has also been drawn in [15] but with a different mathematical approach. But the above results have been contradicted in [16] as no role of Intra Cross Phase Modulation (IXPM), due to inter symbol interference as a result of chromatic dispersion, has been considered. For future work it would be a good task to analyze the behavior of interplay of nonlinear phase noise and chromatic dispersion as well as statistically modeling the NLPN in that case. 6.. CODING NLYSIS For the analysis of symbol error probability of a 16 QM constellation, noise is considered to be Gaussian and having same variance in the two degrees of freedom for all the points in the constellation [17]. But from the analysis of figure (a) and figure (b) we observe that higher signal amplitudes have higher variance of nonlinear phase noise. It clearly indicates that an imbalanced error control coding scheme is required that encodes the data with higher signal amplitudes with more powerful codes than the ones with low signal amplitudes. So an imbalanced but optimized error control coding scheme can further improve the system performance or in other words by optimizing the over head of error control coding scheme the bit error rate can be further reduced. For future it would be an interesting task to optimize the previously proposed error control coding schemes for fiber optical communication in the context of nonlinear phase noise in the system. 30

37 REFERENCES 1. Govind P. grawal,``light Wave Technology Telecommunication Systems, nd Edition, John Wiley & Sons,005.. Keang-Po Ho, ``Phase-Modulated Optical Communication Systems, Springer, Rodger E. Ziemer, Roger L. Peterson, ``Introduction to digital communication, nd Edition, Prentice Hall, pril, J. P. Gordon and L. F. Mollenauer, ``Phase noise in photonic communications systems using linear amplifiers, Optics Letters, Vol. 15, No. 3, pp ,December 1, Keang-Po Ho, Joseph M. Kahn, ``Electronic Compensation Technique to Mitigate Nonlinear Phase Noise, Journal of light wave technology, vol., No. 3, pp. 779, March Keang-Po Ho, ``symptotic probability density of nonlinear phase noise, Optics Letters, Vol. 8, No. 15, pp , ugust 1, Keang-Po Ho, ``Performance Degradation of Phase-Modulated Systems due to Nonlinear Phase Noise, IEEE Photonics Technology Letters, VOL. 15, No. 9, pp , September lan Pak Tao Lau and Joseph M. Kahn, ``Signal Design and Detection in Presence of Nonlinear Phase Noise. Journal of Lightwave Technology, Vol. 5, No. 10, pp , October Ezra Ip, lan Pak Tao Lau, Daniel J. F. Barros and Joseph M. Kahn, ``Compensation of Dispersion and Nonlinear Impairments Using Digital Backpropagation, Journal Of Lightwave Technology, Vol. 6, No. 0,pp , October 15, Per Kylemark, ``Nonlinear Fiber Optical Technologies for Transmission and mplification Technical report - Department of Micro technology and Nano Science, Chalmers University of Technology, Govind P. grawal, ``Nonlinear Fiber Optics, 4th Edition, cademic Press, Ezra Ip and Joseph M. Kahn, ``Increasing Optical Fiber Transmission Capacity beyond Next- Generation Systems, IEEE Lasers and Electro-Optics Society, 008. LEOS st nnual Meeting, pp , November 9-13, Shiva Kumar, ``Effect of dispersion on nonlinear phase noise in optical transmission systems Optics Letters, Vol. 30, No. 4, pp , December 15, G. Green and P. P. Mitra, L. G. L. Wegener, `` Effect of chromatic dispersion on nonlinear phase noise, Optics Letters, Vol. 8, No. 4, pp , December 15,

38 16. Keang-Po Ho, Hsi-Cheng Wang, ``Effect of dispersion on nonlinear phase noise, Optics Letters, Vol. 31, No. 14, pp ,July 15, Erik Strom, ``Notes on Quadrature mplitude Modulation, Lecture notes in digital communications course, course code (SSY-15), Chalmers University of Technology Sweden, November 7,

39 PPENDIX The detailed expression for nonlinear post compensation ( )= + + from [9] is presented in this appendix and it refers to section 4.6 and 5.9 in the main text. x x x x sin cosh cos sinh x sin x sinh x ρsx θ ( ) 4 c r = r r + h( x, ρs ) cosh x cos x cosh x cos x Where ρs = is the OSNR and h ( x, ρ ) s is defined as follows: N σ π 1 x x x sin x sinh x sin x cosh x cos x sinh x h( x, ρ ) tan cot tanh + s = + s 4 s 4 ρ + ρ cosh x + cos x cosh x cos x and xis defined as follows: γ PL x = ρ where P and L are the transmitted powers and system lengths. s 33

40 PPENDIX B The calculations for the radii of modified circular 16 QM constellation are presented in this appendix and they are referred to section 5.11 in the main text. Radius of shell 1 in square 16 QM constellation is given by r 1 = = Radius of shell in square 16 QM constellation is given by Radius of shell 3 in square 16 QM constellation is given by r = = 10 r 3 = = 18 = 3 Difference of radius 1 and radius 3 of square 16 QM constellation = r = r3 r1 = 3 = Radius of 1 st shell of circular 16 QM constellation = r 1' = Radius of nd shell of circular 16 QM constellation = Radius of 3 rd shell of circular 16 QM constellation = Radius of 4 th shell of circular 16 QM constellation = r4' = r3 = 3 r 5 r' = + r1' = + = r 5 7 r3' = + r' = + =